Abstract
In this paper, we discuss a new algorithm to find a non-zero real root
of the transcendental equations using series expansion. This proposed
method is based on the inverse series expansion, which gives a good approximate root than some other existing methods. The implementation
of this algorithm is presented in Matlab and Maple. Sample numerical
examples are presented to illustrate and validate the efficiency of the proposed algorithm. The method will help to implement in the commercial
package for finding a real root of a given transcendental equation.

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Southeast-Asian J. of Sciences, Vol. 7, No. 2 (2019) pp. 106-114
A NEW ALGORITHM FOR COMPUTING A
ROOT OF TRANSCENDENTAL EQUATIONS
USING SERIES EXPANSION
Srinivasarao Thota∗ and Tekle Gemechu
Department of Applied Mathematics
School of Applied Natural Sciences
Adama Science and Technology University
Post Box No. 1888, Adama, Ethiopia
e-mail: srinithota@ymail.com; tekgem@yahoo.com
Abstract
In this paper, we discuss a new algorithm to ﬁnd a non-zero real root
of the transcendental equations using series expansion. This proposed
method is based on the inverse series expansion, which gives a good ap-
proximate root than some other existing methods. The implementation
of this algorithm is presented in Matlab and Maple. Sample numerical
examples are presented to illustrate and validate the eﬃciency of the pro-
posed algorithm. The method will help to implement in the commercial
package for ﬁnding a real root of a given transcendental equation.
1 Introduction
In science, engineering and computing, solving for root or ﬁnding a root of
transcendental equations is playing important role. In the literature, we have
many well known algorithms for root ﬁnding, (for example, Bisection,Muller’s,
Regula-Falsi, Secant, Newton-Raphson methods etc.); the modiﬁed and hybrid
algorithms for ﬁnding an approximate root are available in literature, see for
example [2, 12, 10, 11, 5, 1, 15, 16, 18, 17, 19, 4, 22, 27, 21, 20, 9, 14, 6, 28, 7, 25,
26]. In the present work, we propose a new method based on series expansion,
which ﬁnds quick root in comparison with other existing algorithms. The rest
∗Corresponding Author. Email: srinivasarao.thota@astu.edu.et
Key words: Root ﬁnding algorithms, Transcendental equations, Series expansion, Real root.
2010 AMS Classiﬁcation: 65Hxx, 65H04.
106
107
of paper is arranged as follows: Section 2 describes the proposed method, their
mathematical formulation, calculation steps and ﬂow chart; implementation of
the proposed algorithm in Maple and matlab is presented in Section 3 with
sample computations; and Section 4 discuss some numerical examples to illus-
trate the algorithm and comparisons are made to show eﬃciency of the new
algorithm.
2 Proposed Algorithm
The new iterative formula using series expansion is proposed as
xn+1 = xn
(
xnf
′(xn)
f(xn) + xnf ′(xn)
)
, n = 0, 1, 2, . . . . (1)
By expanding this iterative formula, one can obtain the standard Newton-
Raphson method as in ﬁrst two terms. This is shown in the following theorem.
Theorem 1. Suppose α = 0 is a real exact root of f(x) and θ is a suﬃciently
small neighbourhood of α. Let f ′′(x) exist and f ′(x) = 0 in θ. Then the
iterative formula given in equation (1) produces a sequence of iterations {xn :
n = 0, 1, 2, . . .} with order of convergence p ≥ 2.
Proof. The iterative formula given in equation (1) can be expressed in the
following form
xn+1 = xn
(
xnf
′(xn)
f(xn) + xnf ′(xn)
)
.
Since
lim
xn→α
(
xnf
′(xn)
f(xn) + xnf ′(xn)
)
= 1,
and hence xn+1 = α. This implies that α is a ﬁxed point of the function t(x).
Using the standard expansion of (1− x)−1 as
1
1− x = 1 + x + x
2 + x3 + x4 + · · · (2)
108 A New Algorithm for Computing a root of...
and from equations (1) and (2), we have
xn+1 = xn
(
xnf
′(xn)
f(xn) + xnf ′(xn)
)
= xn
⎛
⎝ 1
1−
(
−f(xn)
xnf ′(xn)
)
⎞
⎠
= xn(1 +
( −f(xn)
xnf ′(xn)
)
+
( −f(xn)
xnf ′(xn)
)2
+
( −f(xn)
xnf ′(xn)
)3
+
( −f(xn)
xnf ′(xn)
)4
+ · · · )
= xn − f(xn)
f ′(xn)
+
1
xn
(
f(xn)
f ′(xn)
)2
− 1
x2n
(
f(xn)
f ′(xn)
)3
+ o
(
1
x3n
(
f(xn)
f ′(xn)
)4)
.
Since f(xn) ≈ 0, when we neglect higher order terms, then the above equa-
tion becomes Newton-Rapson method. Indeed, we have the following formulae
obtained from ﬁrst two, three and four terms of the expansion respectively.
xn+1 = xn − f(xn)
f ′(xn)
.
xn+1 = xn − f(xn)
f ′(xn)
+
1
xn
(
f(xn)
f ′(xn)
)2
.
xn+1 = xn − f(xn)
f ′(xn)
+
1
xn
(
f(xn)
f ′(xn)
)2
− 1
x2n
(
f(xn)
f ′(xn)
)3
.
In the above equations, we obtained Newton-Rapson method having quadratic
convergence in ﬁrst two terms. Therefore, the order of convergence of proposed
methods (1) is at least p ≥ 2.
2.1 Steps for Computing Root
I Select an approximation xn = 0.
II Apply the iterative formula given in equation (1).
III Repeat Step II until we get desired approximate root, for n = 0, 1, 2, . . ..
Flow chat of the proposed algorithm is presented in Figure 1
109
Select Two Approximations of root
If approximate root
satisfy the desired
residual
If approximate root satisfies the
desired residual then
STOP
Compute
Figure 1: Flow chart for proposed algorithm
3 Implementation of Proposed Algorithm
In this section, we discuss the MATLAB and Maple implementation of the
proposed algorithm. The data type SeriesNewton(f,x0,esp,n) describes the
implementation in MATLAB, where f is given non-linear transcendental func-
tion, x0 is the initial approximation of the root, esp is the relative error and n
is the number of iterations required.
function root = SeriesNewton(f,x0,esp,n)
iter = 0; ea = 0; xn = x0;
fd = inline(char(diff(formula(f))),’x’);
disp(’ ----------------------------’);
disp(’ No Root f(Root) %error ’);
disp(’ ---------------------------’);
while (1)
xnold = xn;
xn =x0*(x(0)*fd(x0)/(f(x0)+x0*fd(x0)));
xnnew = xn;
disp(sprintf(’%4d %10.4f %10.2f %8.2f’,iter+1,xn,f(xn),ea));
110 A New Algorithm for Computing a root of...
iter = iter + 1;
if xn = 0, ea = abs((xn - xnold)/xn) * 100; end
x0 = xn;
if ea = n, break, end
end
disp(’ --------------------------------------’);
disp([’Given function f(x) = ’ char(f)]);
disp(sprintf(’Approximate root = %10.10f’,xn));
The following a data type SeriesNewton(f,x0,n) gives the implementation
in Maple, where f is given non-linear transcendental function, x0 is the initial
approximation of the root, and n is the number of iterations required.
SeriesNewton:=proc(f,x0,n)
local iten, fx0;
for iten from 1 by 1 while iten < n+1
do
printf("Iteration %g : ", iten);
x0:=(x0*(x0*subs(x=x0,diff(f,x))/
(subs(x=x0,f)+x0*subs(x=x0,diff(f,x)))));
fx0:=subs(x = x0, f);
end do;
return x0,fx0;
end proc:
Sample computations using the implementation of the proposed algorithm are
presented in Section 4.
4 Numerical Examples
This section provides some numerical examples to discuss the algorithm pre-
sented in Section 2 and comparisons are taken into account to conform that
the algorithm is more eﬃcient than other existing methods.
Example 4.1. Consider the following transcendental equations [11]. We
compare the number of iterations required to get approximation root with accu-
racy of 10−15. The numerical results are provided in Table 1.
a. f(x) = ln(x), with initial approximation 0.5.
b. f(x) = x− esin(x) + 1, with initial approximation 1.5.
c. f(x) = xe−x − 0.1, with initial approximation 0.1.
111
Table 1: Comparing No. of iterations by diﬀerent methods
Fun.
Exact
Root
Regula Falsi
method
Newton Raphson
method
Steﬀen
method
Proposed
method
a. 1.00000 27 Divergent Failure 3
b. 1.69681 & 0 32 Not Convergent Failure 6
c. 0.11183 15 Failure Failure 3
The numerical results given in Table 1 shows that the proposed method is
more eﬃcient than other methods.
Example 4.2. Consider a transcendental equations of the following type.
We ﬁnd approximate root using formulae diﬀerent existing methods to show the
convergence of the proposed algorithm gives a root faster than other methods,
see Table 2. To ﬁnd approximate root, we start with an initial approximation
as 1.5 for Newton-Raphson and Proposed methods; a = 0, b = 1.5 for bisection
and regula-falsi methods. We have an exact real root is 0.5.
f(x) = 2x3 + 11x2 + 12x− 9. (3)
Table 2: Comparing approximate root using proposed algorithm with other
existing methods
Iteration
No.
Newton-Raphson
method
Bisection
methods
Regula-Falsi
method
Proposed
method
1 0.8076923077 0.75 0.2727273 1.026315789
2 0.5428093643 0.375 0.4044266 0.7296759182
3 0.5010101572 0.5625 0.4612480 0.5699486582
4 0.5001005827 0.46875 0.4845290 0.5097474998
5 0.5000105835 0.515625 0.4938624 0.5002347438
6 0.5000005827 0.4921875 0.4975712 0.5000001415
7 0.5000000009 0.5039063 0.4990399 0.4999999998
Example 4.3. This example gives the sample computation using MatLab
and Maple implementation as described in Section 3. Consider a transcendental
equation of the form
f(x) = 2x3 + 11x2 + 12x− 9
112 A New Algorithm for Computing a root of...
with initial approximation of the root as 1.5.
Using MatLab implementation, we have the following computations.
f=inline(’2*x∧3+11*x∧2+12*x-9’,’x’);
function root = SeriesNewton(f,1.5,0.00001,9)
------------------------------------------------------
No Root f(Root) %error
------------------------------------------------------
1 1.026315789 17.0644 ---
2 0.7296759182 6.38981 10.60666586
3 0.5699486582 1.78293 0.4217779832
4 0.5097474998 0.240146 0.0007051480758
5 0.5002347438 0.005752 3.526445668*10−8
6 0.5000001415 3.467e-06 0
7 0.4999999998 -4e-09 0
8 0.4999999996 -7e-09 0
9 0.4999999998 -4e-09 0
-----------------------------------------------------
Given function f(x) = 2*x∧3+11*x∧2+12*x-9
Approximate root = 0.4999999998
Using Maple implementation, we have the following computations.
> f:= 2*x∧3+11*x∧2+12*x-9:
> ExpNewton(f,1.5,9);
Iteration 1 : 1.026315789
Iteration 2 : 0.7296759182
Iteration 3 : 0.5699486582
Iteration 4 : 0.5097474998
Iteration 5 : 0.5002347438
Iteration 6 : 0.5000001415
Iteration 7 : 0.4999999998
Iteration 8 : 0.4999999996
Iteration 9 : 0.4999999998
One can use the implementation of the proposed algorithm to speed up the
manual calculations.
113
5 Conclusion
In this present work, we presented a new algorithm to compute an approxi-
mate real root of a given (nonlinear) transcendental equation. The proposed
new algorithm was based on series expansion. Illustrative examples were given
to prove its better convergence and eﬃciency over some existing methods. The
proposed algorithm is useful for solving some complex real life problems. Im-
plementation of the proposed new algorithm was using Matlab and Maple.
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